\section{Introduction}

Introducing smart phones such as iPhone and Android phones has
boosted up mobile data traffic. According to the
report\cite{cisco_data}, on average, a smart phone generated 24 times
more data traffic than a feature phone. From this mobile data explosion, mobile service
providers are facing system overloading problems. To overcome the
traffic explosion, the providers should enhance their system
capacity. However, the system upgrading has limitations. For instance,
the cost for system upgrading is very expensive. Furthermore, the data
explosion is too rapid for system upgrading to be supported.

Data offloading is an alternative solution to overcome the overloaded
mobile data traffic. As majority of time is spent in indoor area,
offloading mobile data traffic could make service providers support
much more traffic. Therefore, nowadays, femtocells are spreading
rapidly. According to the femtocell forum\cite{femto_forum}, the
number of deployed femto APs is expected to reach 49 million by 2014.
In general, a small base station called femto AP is deployed in a home
or office by manually to build femtocell as shown in
Fig.~\ref{fig:interfemto}. Since the deployment is not organized by
service providers but by users, it should be easy to deploy. Thus,
femto APs have self-configuration
concept\cite{CAG08FNS,HY10CL,JM09,HC09DOF}. Therefore, in most of
service scenarios, the femto AP connects the core network by means of
ISP networks and sets its parameters such as transmission power level
and channel automatically.

However, although being necessary for femtocells, the
self-configuration concepts bring problems. The main issue for the
self-configuration is an interference problem induced from the
transmission power of femto APs. For example, when the power level
is too high, nearby users, who are not served by the femto AP, are
suffered from the interference from the femto AP. On the other hand,
when the power is not enough, the femto AP could not give sufficient
capacity to its serving users.  Thus, finding efficient power control
scheme for femto APs is sophisticated work. There are some previous
works which considered the power control of femto
APs\cite{JM09,HC09DOF,CA09PC}. In the previous works, the distributed
power control schemes control the transmission power to guarantee the
service of macrocell\cite{JM09} or to have designated
coverage\cite{HC09DOF}. In \cite{CA09PC}, game theoretic
approach was exploited to make a distributed power control scheme. The
schemes focused on interference issue or handoff issue. 

% Due to the mobile data explosion, recently, improving capacity of
% wireless links of cellular networks is the major concern of service
% providers\cite{cisco_data}. Especially, the traffic generated from
% indoor areas such as house and office is hard to serve due to path
% loss from walls, althouth users spend large portion of time at indoor
% areas. Furthermore, the increment of capcity requests a lot of money
% to the providers. Therefore, providers start to deploy indoor access
% points such as femto APs to offload the indoor traffic with cheap
% price. Nowdays, femto APs, acting as a small base station, are spreading
% rapidly because of the economical benefits. The number of deployed
% femto APs is expected reaching just under 49 million by
% 2014\cite{femto_forum}.


% Femto APs should be easy to deploy so that users could install the
% femto AP at their house conveniently if they want. Therefore,
% self-configuration is a key feature of femtocell
% networks. 

How to give fair service to users is also important question for
service providers, while, in previous works considering the femto power
control, how to reduce interference or unwilling handoffs was
an important problem. For example, as shown in Fig.~\ref{fig:interfemto}, when each
femtocell has different number of users, femto AP $i$ has more
throughput than  femto AP $j$ to give fair service.  In this paper, a game theoretic approach is exploited to
propose a distributed power control scheme which is power efficient
and fair to users.  In order to consider both fairness and efficiency,
$\alpha$-fairness\cite{MW00FECC} was used to measure network
performance. The $\alpha$-fairness was defined with the $\alpha$-fair
utility function which implicitly indicates the product of fairness
and efficiency\cite{KCS10ATF}.  In our game model, femto APs control
transmission power to maximize their payoff function which is based on
the proportional fairness function which is a $\alpha$-fair utility
function when $\alpha=1$. Our preliminary work was presented in
\cite{ESD09DGT}. In this paper, additionally, we showed both the
uniqueness of Nash Equilibrium (NE) and global stability of the NE.

The rest of this paper is organized as followings. In Section
\ref{sec:system-modelling}, a system model is described and
assumptions are introduced. A game model is suggested in Section
\ref{sec:game-theor-form}. By using the game model, a distributed
power control scheme is generated. The numerical results for the
scheme are given in Section \ref{sec:numerical-example}. In Section
\ref{sec:conclusions}, we conclude the paper.
%기존연구....
%

% Because of the random deployment of femto APs, however, it is hard
% to know neighbor APs which could be interfered from the femto AP. 

%Both efficiency and fairness are important for designing power control scheme.
%In general, fairness measures, which are  defined by various ways, have highest value when the system has the same magnitude for all elements\cite{KCS10ATF}. However, this kind of fairness measure might not suitable for network research, since they do not consider network efficiency. In order to consider both fairness and efficiency, $\alpha$-fairness was introduced\cite{MW00FECC}. The  $\alpha$-fairness is defined with the $\alpha$-fair utility function which implicitly indicates the product of fairness and efficiency. Thus, the $\alpha$-fair utility function is a good measure for network performance. 
\begin{figure}[t]
  \centering
  \includegraphics[width=0.75\textwidth]{fig/femtointermodel}
  \caption{A cellular network overlapped multiple femtocells}
  \label{fig:interfemto}
\end{figure}


\section{System Modeling}
\label{sec:system-modelling}

This paper focuses on scenarios that femto APs are deployed in private
indoor area such as houses and apartments.  In these scenarios, since
houses are partitioned by walls or floor, in general, users are served
by femto APs only when they are in their house. Thus, the number of
serving users is not changed easily and it is unusual that a user
selects a femto AP which is deployed in his neighbor house. For that
reason, we assume that each femto AP knows the number of their serving
users and users are served by femto APs only if they are in their
home.

Furthermore, for the sake of simplicity, we will assume followings to
calculate the throughput of each user.  At first, we assume that every
user has sufficient data to transmit, and each user
gets the same throughput during the service time by a fair
scheduler. We also assume that users feel the identical interference
level when they are in the same house. However, these assumptions do
not seem to be unrealistic due to following reasons: 1) in practical,
the traffic demand of wireless networks increases dramatically and
becomes to be saturated to the network capacity\cite{cisco_data}, 2)
in view of long term average, due to the mobility of users, they have almost
identical channel environment in the same house.

Let $\mathcal{F}$ denote a set of femto APs and $N_i$ denote the
number of users served from the femto AP $i \in \mathcal{F}$. For
notational convenience, in this paper, the set of femto AP is
represented as integer set $\{1,2,\cdots,|\mathcal{F}|\}$, where $|\mathcal{F}|$
denotes the cardinality of $\mathcal{F}$. As shown in
Fig.~\ref{fig:interfemto}, the pathloss between femto AP $i$ and femto
AP $j$ is denoted by $h_{ij}$ or $h_{ji}$, where the channel is
symmetric (i.e., $h_{ij}=h_{ji}$).
Notations, which are used in
this paper, are described in Table~\ref{notations}. 

 Then, the receiving data rate
for users, who are served from femto AP $i \in \mathcal{F}$, can be
expressed as follows :
%==============================================================
\begin{eqnarray}
R_{i} &=& \frac{1}{N_{i}}log(1+\frac{h_{ii}p_i}{I + \sum_{j\neq i}
h_{ij}p_j}),
\end{eqnarray}
%==============================================================
where $I$ and $p_i$
represent noise power plus interference from the macro BS and the
transmission power of femto AP $i\in \mathcal{F}$, respectively. Note
that the transmission power of femto APs are closely related to the
performance of other femto APs.  Since our object is enhancement
of fairness with low transmission power, we define a system performance
measure as a function of transmission power vector $U({\bf p})$ as follows:
%==============================================================
\begin{eqnarray}
U({\bf p}) &=& \sum_{i \in \mathcal{F}} N_{i} \log (R_i)-\beta \sum_{i
  \in \mathcal{F}}  p_i, \label{eq:1}
\end{eqnarray}
%==============================================================
where {\bf p} denotes a power vector $[p_1 , p_2 , \cdots ,
p_{|\mathcal{F}|}]$. $\sum_{i \in \mathcal{F}} N_{i} \log (R_i)$ indicates the proportional fair
function and power consumption is also considered in the subtract term $\beta \sum_{i
  \in \mathcal{F}}  p_i.$


\begin{table}[t!]
  \caption{Summary of Major Notation}
  \begin{centering}
    \begin{tabular}{|c|c|c|}
      \hline
      Notation & Description\tabularnewline
      \hline
      \hline
   $\mathcal{F}$ & set of femto APs \tabularnewline
\hline
 $N_i$ & number of users in $i$-th femto AP\tabularnewline
\hline
 $p_i$ & transmission power of $i$-th femto AP\tabularnewline
\hline
 $U_i(\cdot)$ & payoff function of $i$-th femto AP\tabularnewline
\hline
 $B_i(\cdot)$ & best response function of $i$-th femto AP\tabularnewline
\hline
$p_{-i}$ & power vector except $i$ ($[p_1 , \cdots ,p_{i-1},p_{i+1},\cdots, p_{|\mathcal{F}|}]$)\tabularnewline
    \hline
${\bf p}$ & power vector ($[p_1 , p_2 , \cdots , p_{|\mathcal{F}|}]$)\tabularnewline
    \hline
${\bf B}({\bf p})$ & best response vector ($[B_{1} , B_{2} , \cdots , B_{|\mathcal{F}|}]$) for power vector ${\bf p}$ \tabularnewline
    \hline
\end{tabular}\label{notations}
    \par\end{centering}
\end{table}



\section{Game Theoretic Formulation} 
\label{sec:game-theor-form}

In this section, we will suggest a game model which considers fairness
and the power consumption of femto APs. Moreover, proofs
of the uniqueness and convergence of NE will be provided.

\subsection{Game model}

A game model consists of {\bf players}, {\bf strategy sets} of players,
and {\bf payoff functions} for each player. In this game model, since
the femto APs control their transmission power to enhance proportional
fairness, the {\bf players} and {\bf strategy sets} should be femto APs
and the range of transmission power, respectively. Remaining element
for this game is the payoff function. 

Our goal is making the payoff function so that the equilibrium of this
game converges to the maximum point of Eq.~\eqref{eq:1}.  However,
finding the payoff function is not simple, and, in addition,
additional signal exchange might be needed. Thus, in this work,
heuristic game model is used. 
From Eq.~\eqref{eq:1}, we set the \textbf{payoff function} $U_{i}$ for every
player $i \in \mathcal{F}$ as following
%==============================================================
\begin{eqnarray}
U_{i}({\bf p})   &=&N_i \log(R_i)-\beta p_i\\
  &=& N_{i} \log(\log(1+\frac{h_{ii}p_i}{I + \sum_{j\neq i}
  h_{ij}p_j}))-N_{i}\log(N_{i})-\beta p_i. \label{eq:2}
\end{eqnarray}
%==============================================================
Note that our object function is $U({\bf p}) = \sum_{i \in
  \mathcal{F}}U_{i}({\bf p})$. Thus, when each femto AP tries to
increase their payoff function, the object function might be enhanced. Although the $R_i$ is coupled with other femto APs' transmission
power, each femto AP could calculate his payoff without any message
passing since the denominator $I + \sum_{j\neq i} h_{ij}P_j$ of
Eq.~\eqref{eq:2} is just noise interference level of femto AP $i$ and
the other terms, $N_i$ and $p_i$, are known. Therefore, in totally
distributed manner, by using this game model, each femto AP could
update his transmission power level to maximize his payoff value. The
proposed game model is summarized as following: 
\begin{separation}
{\it Game model : femto AP power game}
\begin{compactenum}[\em i)]  
    \item \textbf{player set :} the set of femto APs $\mathcal{F}$
    \item \textbf{strategy set of player $i$ :}  transmission power
      range $[p_{min} , p_{max}]$
    \item \textbf{payoff of player $i$ :} $N_{i} \log(\log(1+\frac{h_{ii}p_i}{I + \sum_{j\neq i}
  h_{ij}p_j}))-N_{i}\log(N_{i})-\beta p_i$
\end{compactenum}  
\end{separation}
where
$p_{min}$ and $p_{max}$ are the minimum and maximum available power
level for femto APs, repsectively.


%========================================
% \begin{table}
% \caption{Game model}
%   \centering
%   \begin{tabular}{|c|c|}
%     \hline
%     \textbf{player set}& the set of femto APs $\mathcal{F}$  \tabularnewline
%     \hline
%     \textbf{strategy set of player $i$} & transmission power range $[p_{min} , p_{max}]$   \tabularnewline
% \hline
% \textbf{payoff of player $i$} & $N_{i} \log(\log(1+\frac{h_{ii}p_i}{I + \sum_{j\neq i}
%   h_{ij}p_j}))-N_{i}\log(N_{i})-\beta p_i$   \tabularnewline
% \hline
%   \end{tabular}
  
%   \label{tab:gamemodel}
% \end{table}
%========================================

\subsection{Supermodular Game} %\label{subsection1}

Since supermodular game has some nice characteristics such as the
convergence of NE, the game was applied to implement power control
algorithms in some previous works. Thus, it is very useful to show
supermodularity of the proposed game. In this paper, at first, we will
verify that the proposed game is supermodular and, by using the
property of supermodular game that NE is globally stable if the NE is
unique in supermodular game, convergence of the proposed distributed
power control scheme is proved. Supermodular game is defined as
followings.

\begin{separation}
{\it Definition 1 : Supermodular Game\cite{T98SC}}

Let $A_i$ and $\pi_i$ indicate a strategy set and payoff of player
$i$, respectively, where $\mathcal{N}:=\{1, ... , n\}$ denotes a set
of players. Let $A_{-i}$ denote a strategy vector set for every player
except player $i$.  The game is
supermodular if following conditions are satisfied.
\begin{compactenum}[\em i)]
\item For each $i$, $A_i$ is a compact subset of $\mathbb{R}$ which
represents the set of all real numbers.
\item $\pi_i$ is upper semi-continuous in $a_i,a_{-i}$, where $a_i \in
  A_i$ and $a_{-i} \in A_{-i}$. 
\item $\pi_i$ displays increasing differences in
($a_i,a_{-i}$) for all $a_i \in A_i$ and $a_{-i} \in A_{-i}$.  If $\pi_i$ is twice continuously
differentiable with $\frac{\partial^{2} \pi_i}{\partial
a_{i}\partial a_{j}}>0$ for all $i\neq j$, then $a_i$ displays
increasing differences in ($a_i,a_{-i}$). 
 
\end{compactenum}
% \vskip1mm
\end{separation}


Now, we will verify whether the proposed game is supermodular game or
not.  In case of supermodular game, the following three properties
 should be satisfied. 
\begin{compactenum}[\em i)]
\item $[p_{min}, p_{max}]$ is a compact subset of $\mathbb{R}$.

\item For all $p_{i} \in [p_{min}, p_{max}]$, the payoff function
  $U_{i}({\bf p})$
is continuous.

\item $\frac{\partial^{2} U({\bf p})}{\partial p_{i}\partial
p_{j}}>0$ for all $i,j \in \mathcal{F}$
\end{compactenum}
Fortunately, the proposed game has above characteristics, which
implies that the game is supermodular.  The first and second
conditions can be easily known. The remaining thing to be checked is
third condition, which represents the increasing difference
property. Let $I_{t}=I+\sum_{j\neq i}h_{ji}P_{j}$, which represents
total interference arriving at $i \in \mathcal{F}$. Then, the partial
differential function of the payoff function can be expressed as
follows :
%==============================================================
\begin{equation}
\frac{\partial U_{i}({\bf p})}{\partial p_i}=\frac{N_{i}h_{ii}}{(I_{t}
+h_{ii}p_{i})log(1+\frac{h_{ii}p_i}{I_{t}})}-\beta
\end{equation}
%==============================================================
Note that the $I_{t}$ is easily measured by $i \in
\mathcal{F}$ and the payoff function for $i$ is
determined by $I_{t}$ and $p_i$. Therefore, the femto AP can find the
maximum point of the payoff function without any message passing with
other femto APs. 

For the sake of notational convenience,  we set
$D=(I_{t}+h_{ii}p_{i})log(1+\frac{h_{ii}p_i}{I_{t}})$. Then, the second
derivative of the payoff function is represented as following:
%==============================================================
\begin{equation}
\frac{\partial^{2} U_{i}({\bf p})}{\partial p_{i}\partial p_{j}}=\frac{-
N_{i}h_{ii}h_{ji}\{ log(1+\frac{h_{ii}p_i}{I_{t}}) -
\frac{h_{ii}p_{i}}{I_{t}} \}}{D^{2}}.
\end{equation}
%==============================================================
Due to the fact that $\log(1+x)<x$ for all $x>0$
, we can confirm $\frac{\partial^{2}
U_{i}({\bf p})}{\partial p_{i}\partial p_{j}}>0$ because of $\frac{h_{ii}p_{i}}{A}>0$. Therefore, we can
conclude that \textbf{the proposed game is
supermodular game}.

% This type of structure has several remarkable
%properties. The most noticeable property that can be applied to
%decentralized power control algorithm is that the unique NE of the
%supermodular game is globally stable.


\subsection{Nash Equilibrium}

In this subsection, uniqueness of NE will be shown. Since the
uniqueness gives globally stable property in supermodular game, the
power control scheme based on proposed game guarantees convergence and
stability. All proofs are presented in Appendix.

Every user updates his action to maximize payoff, which is called best
response update. The best response could have more than one action,
since there could be multiple maximum points in case of the payoff function. In
this case, users have multiple choices. On the other hand, in a
special case where the best
response set has unique element for every case, users do deterministic
action when they act to maximize their payoff.  The first lemma asserts that the best response
sets for all users have a unique action. Thus, the best response can
be represented as a function of power vector ${\bf p}$.
\begin{lemma} 
Best response of $i$-th femto AP, which is expressed as 
\begin{eqnarray}B_{i}(p_{-i})&=& \arg \max_{p_i \in [0,P_{max}]}
  U_{i}(p_i , p_{-i}),\end{eqnarray}
has a unique element for
every feasible $p_{-i}$. \label{lem:br} 
\end{lemma}
\begin{proof}
Please see Appendix.
\end{proof}

Let ${\bf B}({\bf p})$ denote a best response vector for ${\bf{p}}$,
whose $i$-th element is $B_{i}(p_{-i})$. At each frame, the power levels are changed
according to ${\bf B}({\bf p})$. Lemma \ref{lem:scal} states an
important property called {\em Scalability}, which guarantees
the convergence and stability of the best response based
update\cite{Y95FUPC}. The {\em Scalability} is strongly related with supermodularity.
\begin{lemma}
(Scalability) For all $\alpha > 1$, $\alpha {\bf B}({\bf p}) > {\bf B}(\alpha {\bf p})$. \label{lem:scal}
\end{lemma}
\begin{proof}
Please see Appendix.
\end{proof}

With above lemmas, we conclude that the proposed game has a unique NE
which is globally stable. Uniqueness of NE implies that there are only
one convergence point. Moreover, the \textit{globally stable} property
guarantees that the system converges to the NE from any initial
state. Thus, under the proposed game model, the femtocells are always
operated in the unique NE.
\begin{theorem} \label{thm:uni}
The proportional fair power control game has a unique NE and converges to the NE for any initial power vector.
\end{theorem}
\begin{proof}
Please see Appendix.
\end{proof}

\section{Numerical Example}
\label{sec:numerical-example}

We have shown two important properties of the proposed game, which are
the uniqueness of NE and the globally stable property. Thus, for every
scenario, the proposed game guarantees a deterministic result. 
To verify the performance of the proposed scheme, a conventional
scheme is exploited as a competitive scheme.  \textit{In case of the
\textbf{conventional scheme}, femto APs set their transmission power
level to support target SINR.} For example, when the target SINR of a
femto AP is 10 and the total interference arriving at the femto AP is
0.1, the femto AP sets transmission power level to 1. In this
numerical analysis,
30dB and 40dB is exploited for the target SINR\cite{YS10}.

In our simulations, the interference from neighbor macrocells is
ignored for the sake of simplicity. But, when neighbor macro BSs use
different channel, which means that reuse factor is not 1, the
intercell interference is ignorable. For single macrocell environment,
we assume that femtocells are uniformly distributed randomly in the
macrocell area. For this environment, proportional fair values of
femto users are measured with varying the number of femtocells. The
number of users of each femtocell is assumed to be one or two with
equal probability. For more detail simulation parameters and channel models refer to
\cite{2010arXiv1009.3522J}. From the simulation, we get the following messages.

%==============================================================
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig/profair}
\caption{Proportional fair value vs number of femto cells }\label{fig:proportionalfair}
\end{center}
\end{figure}
%=============================================================
%==============================================================
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig/tranpower}
\caption{Power consumption vs number of femto cells }\label{fig:betapower}
\end{center}
\end{figure}
%=============================================================
Fig.~\ref{fig:proportionalfair} and Fig.~\ref{fig:betapower} show the
proportional fair value and the average power consumption of the femto
APs with varying the number of femto cells, respectively. In the
figures, \textbf{the proposed game model, compared to the conventional
scheme, has slightly better proportional fair value with much less
power consumption}. In Fig ~\ref{fig:proportionalfair}, the proposed
scheme in cases $\beta = 4$ and $\beta = 16$ overcomes the
conventional scheme in cases of that the target SINRs are 40dB and
30dB, respectively, although proposed scheme spends much less
power. The main reason of the improvement is that the proposed game
model is designed so that the femtocells, which have more serving
users, might use more power.

Moreover, \textbf{the proposed scheme is more stable than the
conventional scheme}. As the number of femto cells increases, the
average power consumption of femto APs increases almost linearly in
case of the conventional scheme, while the average transmission power
of the proposed scheme is almost flat. For instance, in the case when
the target SINR is 40dB, the average power is changed from 0.44 to
0.75, whereas the variation of the transmission power of the proposed
scheme when $\beta$ is 4 is changed just from 0.43 to 0.5. Therefore, when a
system operator wants to set the average power consumption of femto
APs to a target level, in case of the proposed scheme, the target
level could be easily achieved.
 
Fig.~\ref{fig:targetpower} shows that the proposed scheme is better
than the conventional scheme in point of total network capacity, which
means that the sum of capacity over all femtocells, although the
conventional scheme spends more transmission power than the proposed
scheme. It is because, under the conventional scheme, in the areas
where femto APs are dense, the femto APs accelerate power consumption
each other in order to achieve the target SINR, while the proposed
scheme restrains the power consumption in the dense areas thanks to
the utility function of the proposed game model in which the best response is
concave function of interference level. Thus, the proposed scheme use power
more efficiently in view of total network capacity.

%==============================================================
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig/capacity}
\caption{Total network capacity vs number of femto cells }\label{fig:targetpower}
\end{center}
\end{figure}
%=============================================================


\section{Conclusions}
\label{sec:conclusions}
Since the femto APs are deployed in distributed manner,
self-organization methods are needed in the femtocell topology.  For
example, the transmission power level should be determined without any
message passing among the APs.  In this paper, we have proposed a
distributed power control method that consider fairness. The scheme is
based on a game model in which the payoff function is designed as a
proportional fairness function. In addition, the payoff function has
transmission power term to restrict the power consumption of the femto
AP. And we have shown
following important properties which guarantee uniqueness of NE and
stability of the NE.
\begin{compactenum}[\em i)]
\item The proposed game is supermodular game, which implies that the proposed game has
NEs.
\item There is unique NE and the NE is globally stable, which is
proved by {\em Scalability} and supermodularity of the game.
\end{compactenum}

Moreover, through the
numerical example, it has been shown that our proposed power
control scheme could be practically implemented in real
environment and could improve fairness among femtocells.


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